Theorem pwpwab 4081

Description: The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
pwpwab	|- ~P ~P A = { x | U. x C_ A }

Distinct variable group:   x, A

Proof of Theorem pwpwab

Step	Hyp	Ref	Expression
1		vex 2818	. . 3 |- x e. _V
2		elpwpw 4080	. . 3 |- ( x e. ~P ~P A <-> ( x e. _V /\ U. x C_ A ) )
3	1, 2	mpbiran 949	. 2 |- ( x e. ~P ~P A <-> U. x C_ A )
4	3	abbi2i 2349	1 |- ~P ~P A = { x | U. x C_ A }

Syntax hints:   = wceq 1398   e. wcel 2205   {cab 2220   _Vcvv 2815   C_ wss 3213   ~Pcpw 3671   U.cuni 3916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-in 3219  df-ss 3226  df-pw 3673  df-uni 3917

This theorem is referenced by:  pwpwssunieq  4082